_ ^ 1 1 + x dx = - Vidyadip

MCQ

$\int_{}^{} {\frac{1}{{\sqrt {1 + \sin x} }}dx} = $

A
$2\sqrt 2 \log \tan \left( {\frac{\pi }{8} + \frac{x}{4}} \right) + c$
B
$\frac{1}{{\sqrt 2 }}\log \tan \left( {\frac{\pi }{8} + \frac{x}{4}} \right) + c$
✓
$\sqrt 2 \log \tan \left( {\frac{\pi }{8} + \frac{x}{4}} \right) + c$
D
$\frac{1}{{2\sqrt 2 }}\log \tan \left( {\frac{\pi }{8} + \frac{x}{4}} \right) + c$

✓

Answer

Correct option: C.

$\sqrt 2 \log \tan \left( {\frac{\pi }{8} + \frac{x}{4}} \right) + c$

c
(c)$\int_{}^{} {\frac{1}{{\sqrt {1 + \sin x} }}} \,dx = \int_{}^{} {\frac{1}{{\sqrt 2 \sin \left( {\frac{\pi }{4} + \frac{x}{2}} \right)}}} \,dx$
$ = \frac{1}{{\sqrt 2 }}\int_{}^{} {{\rm{cosec}}\,\left( {\frac{x}{{\rm{2}}} + \frac{\pi }{4}} \right)} \,dx = \sqrt 2 \log \tan \left( {\frac{\pi }{8} + \frac{x}{4}} \right) + c.$

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